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arXiv 2005.056202020-05math.ATmath.KT

$E_\infty$-胞腔与无限域的一般线性群

$E_\infty$-cells and general linear groups of infinite fields

Soren Galatius, Alexander Kupers, Oscar Randal-Williams

备注：: 82 pages; final version to appear in Duke Mathematical Journal

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AI总结：

本文从$E_\infty$-代数视角研究无限域等的一般线性群，证明其$E_\infty$-同调有斜率2的零化线，分析了线上的群并推导出非稳定同调的若干推论，解答了多位学者的问题。

摘要：

我们从$E_\infty$-代数的视角研究无限域（更一般地，剩余域为无限域的连通半局部环）的一般线性群。我们证明其$E_\infty$-同调存在斜率为2的零化线，并通过确定Steinberg模上所有不变双线性型分析了该线上的群。我们由此推导出一系列关于一般线性群非稳定同调的推论，特别是解答了Rognes、Suslin、Mirzaii等人的问题。

英文摘要：

We study the general linear groups of infinite fields (or more generally connected semi-local rings with infinite residue fields) from the perspective of $E_\infty$-algebras. We prove that there is a vanishing line of slope 2 for their $E_\infty$-homology, and analyse the groups on this line by determining all invariant bilinear forms on Steinberg modules. We deduce from this a number of consequences regarding the unstable homology of general linear groups, in particular answering questions of Rognes, Suslin, Mirzaii, and others.

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